Mathematics 7–10 · Years 7–8
Rolling one die: relative frequency settles towards one sixth
Statistics and probability
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The idea
The relative frequency of a face on a fair die wanders in a short run and settles towards the theoretical probability of 1/6 as the number of rolls grows, and the complement (not a six) settles towards 5/6.
What you need
- 1 standard six-sided die per pair (plastic, 16 mm)
- 1 shaker cup or tray per pair so the die rolls freely
- Tally sheet with columns for faces 1 to 6 and a running total
- Calculator or spreadsheet for the class pool
How to do it
- Write the sample space {1, 2, 3, 4, 5, 6} and the theoretical probability of a six, 1/6, and of not a six, 5/6.
- Predict how many sixes will appear in 60 rolls (10) and how far from 10 you would accept as ordinary.
- Roll the die 60 times, tallying each face. After every 10 rolls record the running relative frequency of a six (sixes so far divided by rolls so far).
- Plot the running relative frequency against the number of rolls on the same axes as the horizontal line at 1/6.
- Pool the class results (for example 10 pairs gives 600 rolls) and compute the pooled relative frequency of each face and of the complement not a six.
- Compare the spread between pairs (60 rolls each) with the pooled figure (600 rolls) and describe what changed.
What you should see
In 60 rolls the count of sixes is 10 on average with a standard deviation of 2.89, so about 78 per cent of pairs land between 7 and 13, and a count of 3 or fewer or of 17 or more happens in about 2 per cent of pairs (binomial, n = 60, p = 1/6). In the class pool of 600 rolls each face is expected 100 times with a standard deviation of 9.13, so pooled counts between 82 and 118 are ordinary. The running relative frequency of a six moves in large steps early (after 10 rolls it can only be 0, 0.1, 0.2 and so on) and drifts within a narrowing band around 0.167 as rolls accumulate. The complement not a six comes out near 0.833 in the pool. The learner knows it worked when at least five of the six pooled face counts sit inside the 82 to 118 band and the pair-level counts scatter more widely than the pooled ones. Each face lands in the band with probability 0.96, but all six do so together in only about 78 classes in 100, while at least five do so in about 96 in 100 (exact multinomial calculation).
What changes
- What you change
- number of rolls
- What you measure
- relative frequency of a six
- What you keep the same
- same die
- same rolling method (shaker cup onto a flat tray)
- every roll counted, none re-rolled
Common misconceptions
Each of these ideas is wrong, and the activity is a chance to test it.
- A six is harder to roll than the other faces.
- After several non-sixes a six is due on the next roll.
- Sixty rolls should give exactly ten of each face if the die is fair.
- A short run that gives 3 sixes in 6 rolls proves the die is biased.
Safety card
Hazards
- small die dropped on the floor is a slip hazard
Controls
- roll inside a tray or shaker cup
- pick up dropped dice at once
Note
No chemicals or heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; ordinary classroom supervision.
Curriculum references
The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.
- Mathematics K–10 Syllabus (2022), Stage 4 Probability; page read 2026-09-22MA4-PRO-C-01
- Australian Curriculum v9AC9M7P01AC9M7P02AC9M8P01
Sources
The pages the author read to write this activity.